Using piecewise-linear reconstruction to constructing a low-dissipation HLL method for numerical solution of hydrodynamics equation

In this paper, one construction of the original «Harten-Lax-van Leer» method using a piecewise-linear reconstruction of physical variables is described. The obtained numerical method makes possible to reproduce a low-dissipation solution at discontinuities. To verify the method, we used the classical problems with an analytical solution based on various configurations of shock waves, contact discontinuities, and rarefaction waves. On the Sod-like problem, the order of accuracy of the developed numerical method was studied, it was shown that the main suppression of the order of accuracy occurs when the rarefaction wave is reproduced. The numerical method was verified by means of a three-dimensional Sedov test of a point explosion, and on the problem of a supernova Ia type explosion with two symmetric ignition points, leading to the formation of a G1.9+0.3 like remnant.